Make public headers self-contained and stop relying on PCH

[TrueBrain]

Summary

Initially I wanted to make an issue first, to talk this over. But code speaks louder than any words, so I ended up making this PR instead. But this PR is much more a "do you want to go in this direction?" question that should have been asked in an issue.
Sorry :(

When compiling this (and many of the other carbon) repositories on Linux, you run into the issue that a bunch of include headers are missing. And this totally make sense, as these repositories use PCH with <windows.h>. Which means on Windows, a ton of system headers are added without people realizing (as windows.h is a terrible header in this regard).

Building on MacOS will tell about most: but not all. And this is because GCC has a different opinion about system headers than others do: it minimizes them as much as it can. Famously, since GCC 13 they removed the inclusion of cstdint by their headers. To take this as example, it means:

• On Windows, <windows.h> will transitively include this header.
• On MacOS, the fact you use libc++ will transitively include this header (for now at least).
• On Linux/GCC: it will not.

So we run into compiler-errors on Linux. Fixing this can be done in several ways, but it also highlights a bigger problem: something that works on one platform, fails on another. And those can be annoying to deal with after-the-fact.

While delving into this further, I noticed a bit of a more tricky problem to deal with:

No file in the "include" folder includes StdAfx.h, but some do depend on its context via PCH. This means there is a knock-on effect through all repositories that depend on core: they all need a PCH to at least cover the include files in the PCH of core. And that can be unexpected.

So this PR sets out to fix these issues:

• Make public headers self-contained (and validate that with a self-test)
• Make this repository build without PCH

Additionally, I let Claude write some code for the CI, so it covers that non-PCH keeps on building. Just to prevent regression on this front.

After this PR you end up with public-headers that don't implicitly need a PCH to contain some includes. In other words: this fixes that silently the StdAfx.h leaked into the public headers, although StdAfx.h itself wasn't made public. This also means that after this PR PCH becomes a compile-performance optimization again, and not a dependency of its own.

That all said and done, the actual requirement to build on Linux is a much smaller change: https://github.com/carbonengine/core/compare/main...TrueBrain:carbonengine-core:push-xkzzrvtkskxx?expand=1. I am also fine if that is the route to take.

PS: of course if this PR is accepted, I can do the same work on other repositories that have the same issue. About 50% of them currently fail on missing headers, in different degrees.
PPS: the no-PCH build most likely fails on MacOS till #41 lands.

AI assistance disclosure

Claude wrote the code; I instructed what I wanted to achieve. Verified it is matching up with my expectations.

Type of change

• Bug fix
• New feature
• Refactor / cleanup (no behaviour change)
• Documentation
• Build, CI, or tooling
• Breaking change (public API or ABI)
• Other (describe below)

Linked issue (optional)

What changed

• Ensure public headers are including all files they depend on
• Ensure Linux doesn't fail on missing headers (it will still fail on other issues; but that is for other PRs)
• Ensure CI prevents regression

Testing

• Build with PCH on Linux and Windows
• Build without PCH on Linux and Windows

Platforms tested

• Windows
• macOS
• Linux

Screenshots / captures

Checklist

• I've read CONTRIBUTING.md.
• My commits follow the commit-message style described there.
• I've added or updated tests where it made sense.
• I've updated docs / inline API comments for any behaviour change.
• My CLA / ICLA is signed (the bot will let you know if it isn't).

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Full disclosure: I am employed by Fenris Creations, although I have no involvement with the Carbon project. I work on this in my free time under my own name.